Book contents
- Frontmatter
- Contents
- Introduction
- Part Three Metric and topological spaces
- Part Four Functions of a vector variable
- 17 Differentiating functions of a vector variable
- 18 Integrating functions of several variables
- 19 Differential manifolds in Euclidean space
- Appendix B Linear algebra
- Appendix C Exterior algebras and the cross product
- Appendix D Tychonoff's theorem
- Index
- Contents for Volume I
- Contents for Volume III
17 - Differentiating functions of a vector variable
from Part Four - Functions of a vector variable
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Introduction
- Part Three Metric and topological spaces
- Part Four Functions of a vector variable
- 17 Differentiating functions of a vector variable
- 18 Integrating functions of several variables
- 19 Differential manifolds in Euclidean space
- Appendix B Linear algebra
- Appendix C Exterior algebras and the cross product
- Appendix D Tychonoff's theorem
- Index
- Contents for Volume I
- Contents for Volume III
Summary
Differentiating functions of a vector variable
In Part Two, we considered continuity and limiting properties of real-valued functions of a real variable – functions defined on a subset of R. In Part Three we extended these ideas to functions between metric spaces, or between topological spaces. In particular, these results apply to functions of several real variables – functions defined on a subset of Rd.
We now turn to differentiation. This involves linearity: we therefore consider functions defined on a subset U of a real normed space (E, ∥.∥E) taking values in a real normed space (F, ∥.∥F). In fact, our principal concern will be with functions of several real variables (functions defined on an open subset of Rd), but it is worth proceeding in a more general way. First, this illustrates more clearly the basic ideas that lie behind the theory. Secondly, even in the case where we consider functions defined on a finite-dimensional Euclidean space, there are advantages in proceeding in a coordinate free way; not only is the notation simpler, but also the results are seen to be independent of any particular choice of coordinates.
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- Information
- A Course in Mathematical Analysis , pp. 485 - 512Publisher: Cambridge University PressPrint publication year: 2014